Expert: Sherry Wallin - 2/16/2009

Question
(A x B)intersect(B x C) = (A intersect C)x(B intersect D)

i don't know where to start, all i know is that if x is an element of AxB intersect BxC then its in AxB and BxC, but how do I write a proof, can you walk me through it? thank you!

Answer
Hey Dave~
   I think you have written the problem wrong. Correct me if I am wrong but I think you mean:
(A x B)intersect(C x D) = (A intersect C)x(B intersect D
'/\' can be used for intersection

In any case I can't answer the question the way you have it written because I don't believe it is true.

2nd Point: any time you have to prove set equality this is an if and only if type proof. You assume x is in (A x B)/\(C x D) and show x is in (A/\C)x(B/\D). Then you assume x is in(A/\C)x(B/\D) and show x is in (A x B)/\(C x D. In other words this is a bi-conditional proof (<->).

I will show you how to prove one direction based on how I believe the question was suppose to be written:

If x is in (AxB)/\(BxC) then let's let x = (a,b) where (a,b) is in (AxB) and (a,b) is in (CxD) which implies that a is in A and a is in C while b is in B and b is in D. Since both A and C contain a, a is in
(A/\C), similarly both B and D contain b so b is in (B/\D), thus (a,b) is in (A/\C)x(B/\D). Now you need to assume x = (a,b) is in
(A/\C)x(B/\D) and show it is in (AxB)/\(BxC) to complete the proof. It is almost the first direction going backwards actually.

Let me know how well you understand.

Math Prof